Problem: Given two positive numbers a,b such that a<b. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
Answer Choices:
A. ab(b+a)2​
B. 8b(b+a)2​
C. ab(b−a)2​
D. 8a(b−a)2​
E. 8b(b−a)2​
Solution:
We first prove that the A.M., 2b+a​, is greater than the G.M., ab​,b>a.